Normally, with the breaking of the time diffeomorphism, extra degrees of freedom are often added on top of two tensorial degree of freedom in GR. In particular, the spatial covariant gravity which propagate up to 3 dynamical degrees of freedom has been explored extensively [1]}. It is also shown that under certain conditions, the spatial covariant gravities can have just two tensorial degrees of freedom and no propagating scalar mode [2]}. In [3]}, [4]}, the above action has also been extended by introducing \(\dot{N}\)  in the Lagrangian through \(\frac{1}{N}(\dot{N} - N^i \nabla _i N)\) . Since such terms does not contribute to the GWs at quadratic order, we will not consider it in this paper.
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